Papers by Tomoyuki Takenawa
Nonlinearity, Jan 13, 2003
We apply the algebraic-geometric techniques developed for the study of mappings which have the si... more We apply the algebraic-geometric techniques developed for the study of mappings which have the singularity confinement property to mappings which are integrable through linearisation. The main difference with respect to the previous studies is that the linearisable mappings have generically unconfined singularities. Despite this fact we are able to provide a complete description of the dynamics of these mappings and derive rigorously their growth properties.
arXiv (Cornell University), Apr 30, 2002
We apply the algebraic-geometric techniques developed for the study of mappings which have the si... more We apply the algebraic-geometric techniques developed for the study of mappings which have the singularity confinement property to mappings which are integrable through linearisation. The main difference with respect to the previous studies is that the linearisable mappings have generically unconfined singularities. Despite this fact we are able to provide a complete description of the dynamics of these mappings and derive rigorously their growth properties.
Springer eBooks, 2020
In a prior paper the authors obtained a four-dimensional discrete integrable dynamical system by ... more In a prior paper the authors obtained a four-dimensional discrete integrable dynamical system by the traveling wave reduction from the lattice super-KdV equation in a case of finitely generated Grassmann algebra. The system is a coupling of a Quispel-Roberts-Thompson map and a linear map but does not satisfy the singularity confinement criterion. It was conjectured that the dynamical degree of this system grows quadratically. In this paper, constructing a rational variety where the system is lifted to an algebraically stable map and using the action of the map on the Picard lattice, we prove this conjecture. We also show that invariants can be found through the same technique.
PMNP 2017: 50 years of IST, Apr 13, 2017
arXiv: Dynamical Systems, 2019
In a prior paper the authors obtained a four-dimensional discrete integrable dynamical system by ... more In a prior paper the authors obtained a four-dimensional discrete integrable dynamical system by the traveling wave reduction from the lattice super-KdV equation in a case of finitely generated Grassmann algebra. The system is a coupling of a Quispel-Roberts-Thompson map and a linear map but does not satisfy the singularity confinement criterion. It was conjectured that the dynamical degree of this system grows quadratically. In this paper, constructing a rational variety where the system is lifted to an algebraically stable map and using the action of the map on the Picard lattice, we prove this conjecture. We also show that invariants can be found through the same technique.
Journal of Physics A: Mathematical and Theoretical, 2019
A geometric study of two 4-dimensional mappings is given. By the resolution of indeterminacy they... more A geometric study of two 4-dimensional mappings is given. By the resolution of indeterminacy they are lifted to pseudo-automorphisms of rational varieties obtained from (P 1 ) 4 by blowing-up along sixteen 2-dimensional subvarieties. The symmetry groups, the invariants and the degree growth rates are computed from the linearisation on the corresponding Néron-Severi bilattices. It turns out that the deautonomised version of one of the mappings is a Bäcklund transformation of a direct product of the fourth Painlevé equation which has 2 type affine Weyl group symmetry, while that of the other mapping is of Noumi-Yamada's A (1) 5 Painlevé equation.
Symmetry, Integrability and Geometry: Methods and Applications, 2018
Although the theory of discrete Painlevé (dP) equations is rather young, more and more examples o... more Although the theory of discrete Painlevé (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to recognize these equations, to be able to identify their type, and to see where they belong in the classification scheme. The definite classification scheme for dP equations was proposed by H. Sakai, who used geometric ideas to identify 22 different classes of these equations. However, in a major contrast with the theory of ordinary differential Painlevé equations, there are infinitely many non-equivalent discrete equations in each class. Thus, there is no general form for a dP equation in each class, although some nice canonical examples in each equation class are known. The main objective of this paper is to illustrate that, in addition to providing the classification scheme, the geometric ideas of Sakai give us a powerful tool to study dP equations. We consider a very complicated example of a dP equation that describes a simple Schlesinger transformation of a Fuchsian system and we show how this equation can be identified with a much simpler canonical example of the dP equation of the same type and moreover, we give an explicit change of coordinates transforming one equation into the other. Among our main tools are the birational representation of the affine Weyl symmetry group of the equation and the period map. Even though we focus on a concrete example, the techniques that we use are general and can be easily adapted to other examples.
Journal of Physics A: Mathematical and Theoretical, 2017
It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the... more It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painlevé equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painlevé equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.
Contemporary Mathematics, 2015
We present two examples of reductions from the evolution equations describing discrete Schlesinge... more We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painlevé equations: difference Painlevé equation d-P A (1) * 2 with the symmetry group E (1) 6 and difference Painlevé equation d-P A (1) * 1 with the symmetry group E (1) 7 . In both cases we describe in detail how to compute their Okamoto space of the initial conditions and emphasize the role played by geometry in helping us to understand the structure of the reduction, a choice of a good coordinate system describing the equation, and how to compare it with other instances of equations of the same type.
Some Weyl group acts on a family of rational varieties obtained by successive blowups at m (m ≥ n... more Some Weyl group acts on a family of rational varieties obtained by successive blowups at m (m ≥ n + 2) points in the projective space P n (C). In this paper we study the case where all the points of blow-ups lie on a certain elliptic curve in P n . Investigating the action of Weyl group on the Picard groups on the elliptic curve and on rational varieties, we show that the action on the parameters can be written as a group of linear transformations on the (m + 1)-st power of a torus.
SSRN Electronic Journal, 2013
We present examples of a parameterized optimization problem, with a continuous objective function... more We present examples of a parameterized optimization problem, with a continuous objective function differentiable with respect to the parameter, that admits a unique optimal solution, but whose optimal value function is not differentiable. We also show independence of Danskin's and Milgrom and Segal's envelope theorems. Journal of Economic Literature Classification Number: C61.
We prove that the general isolevel set of the ultra-discrete periodic Toda lattice is isomorphic ... more We prove that the general isolevel set of the ultra-discrete periodic Toda lattice is isomorphic to the tropical Jacobian associated with the tropical spectral curve. This result implies that the theta function solution obtained in the authors' previous paper is the complete solution. We also propose a method to solve the initial value problem.
Nonlinearity, 2003
We apply the algebraic-geometric techniques developed for the study of mappings which have the si... more We apply the algebraic-geometric techniques developed for the study of mappings which have the singularity confinement property to mappings which are integrable through linearisation. The main difference with respect to the previous studies is that the linearisable mappings have generically unconfined singularities. Despite this fact we are able to provide a complete description of the dynamics of these mappings and derive rigorously their growth properties.
Letters in Mathematical Physics, 2007
ABSTRACT
Journal of Physics A: Mathematical and Theoretical, 2012
We classify two dimensional integrable mappings by investigating the actions on the fiber space o... more We classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces. While the QRT mappings can be restricted on each fiber, there exist several classes of integrable mappings which exchange fibers. We also show an equivalent condition when a generalized Halphen surface becomes a Halphen surface of index m.
Journal of Nonlinear Mathematical Physics, 2021
In many cases rational surfaces obtained by desingularization of birational dynamical systems are... more In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this method to the study of various integrable or linearizable mappings, including discrete versions of reduced Nahm equations.
Japan Journal of Industrial and Applied Mathematics, 2013
We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algor... more We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial optimization problems with polynomial equality constraints can be modified equivalently so that the associated semidefinite programming relaxation problems have no duality gap. Elementary proofs for some criteria on reality of ideals are also given.
International Mathematics Research Notices, 2010
We propose a method to study the integrable cellular automata with periodic boundary conditions, ... more We propose a method to study the integrable cellular automata with periodic boundary conditions, via the tropical spectral curve and its Jacobian. We introduce the tropical version of eigenvector map from the isolevel set to a divisor class on the tropical hyperelliptic curve. We also provide some conjectures related to the divisor class and the Jacobian. Finally, we apply our method to the periodic box and ball system and clarify the algebro-geometrical meaning of the real torus introduced for its initial value problem.
Communications in Mathematical Physics, 2009
We introduce a tropical analogue of Fay's trisecant identity for a special family of hyperellipti... more We introduce a tropical analogue of Fay's trisecant identity for a special family of hyperelliptic tropical curves. We apply it to obtain the general solution of the ultra-discrete Toda lattice with periodic boundary conditions in terms of the tropical Riemann's theta function.
Advances in Mathematics, 2009
Starting from certain rational varieties blown-up from (P 1 ) N , we construct a tropical, i.e., ... more Starting from certain rational varieties blown-up from (P 1 ) N , we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo isomorphisms of the varieties. Furthermore, we introduce a geometric framework of τ-functions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type, our construction yields (higher order) q-difference Painlevé equations.
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Papers by Tomoyuki Takenawa